That story of the young Athanasius, whether apocryphal or not, illutrates the difference that deLugo is incapable of comprehending. If the story occurred, the bishop was incorrect, that the young Athanasius play-baptizing someone would have validly baptized. There's no need to dismiss the story as apocryphal simply because a bishop may have had a wrong opinion. St. Cyprian had an opinion about re-baptism that was later declared heretical. Just because this was some bishop from the time of the Fathers doesn't make his opinion correct.
So according to the external intention opinion, the play-baptism of young Athanasius would have been a valid Baptism.
But, while performing the rite exteriorly, he had no intention to actually DO what the Church DOES, i.e. to perform the Rite of Baptism.
On the ther extreme, promoted by the ignorant and mendacious De Lugo, you have to internally INTEND what the Church INTENDS. That's preposterous. Otherwise, that classic example of an atheist being able to baptize would be absurd.
What has to happen is that someone needs to internally intend to DO what the Church DOES, i.e. to perform the Catholic Rite of Ordination / Consecration / Mass / Baptism, etc. ... vs. jesting about it, vs. mocking it, vs. being insane or half-asleep, etc. and going through the motions.
There's no difference whatseover between LACKING the internal intention vs. having a contrary internal intention. In both cases, the necessary intention to perform the Sacrament is lacking.
This hypotehtical Mason +Lienart could have wished all he wanted in his head that the Sacrament not be valid, but when he put on his vestments and followed the Catholic Ritual in public during a public celebration of the Rite, he intended internally to DO what the Church DOES, and the Church imposes upon that action the intention that the Church has in having these ministers do it.
But the feeble mind of De Lugo is incapable of comprehending these distinctions. De Lugo confounds internal intention with the internal intention to intend what the Church intends, whereas there must be internal intention to DO what the Church does, thus explaining why an atheist can validly baptize.
It's very straightforward, but has been distorted by the subjectivists. This trend toward subjectivism and relativism has been growing for the past 500 years or so, culminating in the doctrinal relativism of Wojtyla and the moral relativism of Bergoglio.